The study of quantum chaos is a well-developed subject, but the connection to black holes mentioned in the previous section led to a new diagnostic, the out-of-time-order correlator, which can be applied to study chaos in any many-body quantum system [1]}, [2]}. This has led to a number of novel insights. One of these is the idea that there is a characteristic velocity, the “butterfly velocity”, at which chaos propagates through a system. Another is a general bound \(\lambda \le 2\pi T \)  on the growth of chaos, where \(\lambda \)  is a quantum version of a Lyapunov exponent which appears in the out-of-time-order correlator. This bound is saturated by black holes, and seems to hold for rather general quantum systems [3]}. This bound is similar to the “Planckian” bound on many-body transport \(1/\tau \lesssim T \) 
[4]},
which appears to be realized in the linear in \(T\)  resistivity of high temperature superconductors.
